Blow-up Solution for $4$-dimensional Generalized Emden-Fowler Equation with Exponential Nonlinearity

Abstract

Using some nonlinear domain decomposition method, we prove the existence of singular limits for solution of generalized Emden-Fowler equation with exponential nonlinearity in fourth-dimensional given by \[ \begin{cases} \Delta (a(x) \Delta u) - V(x) \operatorname{div}(a(x) \nabla u) = \rho^{4} a(x) e^{u} & \textrm{in $\Omega \subset \mathbb{R}^{4}$}, \\ u = \Delta u = 0 & \textrm{on $\partial \Omega$}. \end{cases} \] The leading part $\Delta$ is, usually, called Laplacian operator. The potential $V(x)$ belongs to $L^{\infty}_{\operatorname{loc}}(\mathbb{R}^{4})$ it is smooth and bounded and $a = a(x)$ is a given smooth function over $\overline{\Omega}$, called the Schrödinger wave function. Namely, we are still looking for solutions which concentrate at the points $x^j \in \Omega$, $j = 1,\ldots,m$ as the parameter $\rho$ tends to $0$. We find sufficient conditions under which, as $\rho$ tend to $0$, there exists an explicit class of solutions which admit a concentration behavior with a prescribed bubble profile around some given $m$-points in $\Omega$, for any given integer $m$. These are the so-called singular limits. The candidate $m$-points of concentration must be nondegenerate (in essential way) critical points of a suitable finite dimensional functional explicitly and the higher order Green's function with respect to the imposed boundary conditions.